Complete reduction bases for the principal induced representations of the crystallographic point groups

An ordered partition P of a point group G is constructed in left cosets H...βα=...BβAαH related to a subgroup H by means of selected genitors A, B, ...: P={H...βαvertical strokeα=1 to a, β=1 to b, ...}, ab... =vertical strokeGvertical stroke/vertical strokeHvertical stroke. This partition P spans a principal induced representation (PIR) R(H:G) of G. Then a basis L of this PIR is built: L={V...kjvertical strokej=1 to a, k=1 to b, ...} with V...kj=...Σbβ=1 Σaα=1 exp[2iπ(...+kβ/b=jα/a)]H...βα. In many cases L is a complete reduction basis (CRB) of R(H:G) for which all matrices are fully reduced. The possibility of obtaining such a CRB depends on the algebraic structure of the group G, on the considered subgroup H and on the choice of genitors A, B, .... Methods are proposed using subgroup chain properties, invariatn inductor subgroup properties, direct product properties etc. These methods have been applied to crystallographic point groups. Complete tables of CRBs are recorded for all PIRs of all crystallographic point groups except for a few PIRs of the point groups 432, anti 43m and manti 3m. (orig.)